Journal of Statistical Physics

, Volume 177, Issue 6, pp 1172–1206 | Cite as

From Coalescing Random Walks on a Torus to Kingman’s Coalescent

  • J. Beltrán
  • E. Chavez
  • C. LandimEmail author


Let \({\mathbb T}^d_N\), \(d\ge 2\), be the discrete d-dimensional torus with \(N^d\) points. Place a particle at each site of \({\mathbb T}^d_N\) and let them evolve as independent, nearest-neighbor, symmetric, continuous-time random walks. Each time two particles meet, they coalesce into one. Denote by \(C_N\) the first time the set of particles is reduced to a singleton. Cox (Ann Probab 17:1333–1366, 1989) proved the existence of a time-scale \(\theta _N\) for which \(C_N/\theta _N\) converges to the sum of independent exponential random variables. Denote by \(Z^N_t\) the total number of particles at time t. We prove that the sequence of Markov chains \((Z^N_{t\theta _N})_{t\ge 0}\) converges to the total number of partitions in Kingman’s coalescent.


Interacting particle systems Martingale problem Markov chain model reduction Kingman’s coalescent 

Mathematics Subject Classification

82C22 60K35 60F99 



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Authors and Affiliations

  1. 1.IMCALimaPeru
  2. 2.PUCPLimaPeru
  3. 3.IMPARio de JaneiroBrasil
  4. 4.CNRS UMR 6085, Université de RouenSaint-Étienne-du-RouvrayFrance

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